With a very strong legacy of finite element data models, in most engineering industries, more and more legacy FEM data is being read in as building blocks for new designs. This is fast changing the engineering design world to an analysis-driven design as opposed to traditional CAD-based design.
In this light, one of the biggest challenges that the designer faces today is to create geometry from legacy FEM data that is partially altered. The first step towards building surfaces or geometry abstractions from the input mesh is to smooth shell meshes in 3D space. Very few 3D smoothers, known today, can meet this challenge. Such a smoother could also be looked at as a preliminary “mesh morphing” technique.
The simplest mesh smoothing technique is Laplacian smoothing where a node is moved to the centroid of its neighboring vertices. This method operates heuristically, and has no control on mesh quality and often throws nodes outside concave domains producing inverted or invalid elements.
Several variants of the Laplacian smoothing technique include “smart” Laplacian methods where element distortion metrices are checked before the node is moved. Length or area weighted Laplacian methods are able to influence repositioning with initial location. In concave domains, however, these techniques still fail to produce stable meshes. Sometimes, area weighted methods help sense inverted elements and can be leveraged by smart Laplacian approaches to prevent moving such nodes. Haber et al. proposed a family of Isoparametric-Laplacian mesh control techniques, meant specially for transfinite meshes. These schemes tend to produce better skewed quadrilaterals but have most of the other Laplacian disadvantages.
In optimization based smoothing, nodes are not moved based on a heuristic algorithm such as Laplacian smoothing, they are moved to minimize a given distortion metric. The distortion metrics are related to the max/min included angles of elements, element skew value, element aspect ratio, element area, element edge length, etc.
One of the early optimization techniques was developed by Parthasarathy and Kodiyalam. They solved a non-linear optimization problem in an effort to repair quad-tree and octree meshes. Shephard and Georges reported similar findings. Amenta et al used linear programming techniques to solve triangular meshes locally. Jacquotte and Coussement developed an optimization based approach for both 2D and 3D structural grids. Freitag et al proposed a local optimization technique for 2D triangular meshes that can serve as the core of an efficient parallel algorithm. Later, Freitag and Oliver-Gooch extended that to 3D grids. These methods produced extremely good quality meshes but involved element repair work like edge-swapping, bad element collapse etc. Moreover the efficiency of these optimization based smoothers are about five to 40 times slower than Laplacian smoothing. Both Freitag and Canann et al later combined several smart Laplacian methods with optimization-based techniques to create hybrid algorithms to improve efficiency. These methods, however, fail to recognize and preserve mapped meshes.
Non-Laplacian, physics-based or non-iterative, direct smoothing algorithms have also emerged in the recent past. Lohner et al used a spring mounted system between nodes to smooth. Shimada, in his bubble packing model, proposed a method where close nodes repel and distant ones attract each other. Bossen and Heckbert proposed an exponential function with similar properties. These models work in 2D space and are not general purpose. A recent direct, non-iterative approach computes artificial stiffness matrices for the mesh to smooth and tries to minimize the strain-energy of the system. While the results are interesting, such methods tend to loose their computational advantage for large models. Tam and Armstrong developed an integer programming based mesh control algorithm for mapped quadrilateral and hexahedral meshes. This method works on a collection of connected subregions and is quite special purpose. Zhou and Shimada have recently proposed an angle smoother in 2D that tends to mount torsion springs between nodes and minimize the system energy. Results prove its merit on certain concave domains and mixed meshes. Winslow smoothing is another efficient technique to reposition nodes of predominantly structured elliptic meshes. Knupp's investigation in this area provides important results. Knupp also proposes “condition numbers” as a good yardstick for measuring mesh quality.
There is a need in the art for a system, process, and computer program product for improved smoothing.